Should I reserve seats on these trains?

As far as I know, there is no combined ticket for express trains. JR provides a Jyoshaken (for a train/distance) and a Tokyuken (for an express/time). All could be bought from the machine. The machine just gives you the needed pieces of tickets.
https://www.westjr.co.jp/global/en/howto/ticket_02.html#fqtop
http://www.jreast.co.jp/e/ticket/examining.html

I usually use a non-reserved seat, but you could specify the exact seat as in the simulator in the page #11 below. It is in Japanese, but just clicks would lead you there from #00. I hope you could take neighboring seats if you push "2 Adults" in #07. All the instructions should be in English as in the Youtube video below.
http://www.jreast.co.jp/mv-guide/demo/shinkansen/11_seatmap.html
http://www.jreast.co.jp/mv-guide/demo/shinkansen/00_menu.html
http://www.jreast.co.jp/mv-guide/demo/shinkansen/07_ninzuu.html
https://www.youtube.com/watch?v=4in8PC6Ezbc

Koyo would have its peak in the later half of Nov in Kanazawa. Kenrokuen is reported 126k visitors in a season.
http://weathernews.jp/koyo/cgi/search_result.fcgi?id=25605

Anyway, I have checked the current seat status (8 Nov) and it seems it is not so crowded now. But it would be better to drop at Midori no Madoguchi or View Plaza once you arrived at Kyoto or Airport.

Kyoto = Kanazawa
12 Nov: No seats(9:09, 19:54, 20:38), Few seats(10:09, 10:40)
19 Nov: Few seats(10:40)
26 Nov: All OK

Kanazawa = Tokyo
14 Nov: All OK
21 Nov: All OK
28 Nov: All OK

http://www.jr.cyberstation.ne.jp/vacancy/Vacancy.html
http://www.jreast.co.jp/e/ticket/purchase.html

But I thought Mt. Fuji was on the left when heading towards Tokyo - when I travelled from Kyoto to Tokyo, I remember sitting on the left to see Fuji.

There's two conversations going on here. Just to be clear:

Mount Fuji will appear on the right side of the train when traveling the Tokaido or Hokuriku Line in the direction of Tokyo.

Mount Fuji will be on the left side of the train when traveling the Tokaido Line in the direction of Tokyo.

That is not any clear. You mention 2 sides/directions for the Tokaido line.

I'm sure when you typed Tokaido and Hokuriku, you meant Chuo and Hokuriku.

Mt. Fuji will be seen from the Tokaido line, on the left when heading towards Tokyo. From the Chuo line, just the top of Mt. Fuji will be seen on the right side. I am afraid that Mt. Fuji would not be seen from the Hokuriku Shinkansen.

For the afternoon train from Kanazawa to Tokyo, I would take the left hand side. If you were on the right side, you would have to close the curtains to block the sunshine.

As of 10:20 on 12 Nov (Saturday), reserved seats both for 10:40 and 11:10 Thunderbird trains from Kyoto to Kanazawa are full. The next 11:40 train has many seats left.

To everyone, and especially LuisJP,
Thank you so very much for your kind inputs!

LuisJP, seriously I can't thank you enough for taking time to write such detailed explanation and provide me with lots of helpful links!
I will make sure to book our Kyoto-Kanazawa and Kanazawa-Tokyo trips as soon as I arrive at KIX. Hopefully there will still be seats :)

One last question regarding luggage. I will need to store my large luggage between 8AM-1PM. If the 24-hr lockers are full and the others are not yet open, what should I do?

Your departing day is Monday, right? So, I do not expect extremely crowded situation for that day. Anyway, I have got many hits for baggage keeping "Nimotsu Azukari" or "荷物預かり". One of the examples is Hokutetsu Kanazawa, at B1F at Kanazawa station, offering such from 6:00.
http://nacchan.net/ishikawa/8210.html

Another site for coin lockers is here.
https://www.google.com/maps/d/viewer?mid=1iQ6c0YLkrJAdRv-5VnippbFQ9yM&...

If you happened to face a problem, just drop at the tourist information center. At Kanazawa station, it opens from 8:30.
http://www.kanazawa-kankoukyoukai.or.jp/access/info_info.php

Or, sending the luggage to your Tokyo Hotel? If the luggage is smaller than (A cm x B cm x C cm, A + B + C is less than 160) and lighter than 30 kg, it would cost 1710 yen by Japan Post. It would be arrived by next morning. Yamato and Sagawa should do similar. You could send the luggage on Sunday, or on Monday after your trial to find coin lockers and such.
http://www.japan-guide.com/e/e2278.html
http://www.post.japanpost.jp/index_en.html

Another idea might be a taxi. You charter a taxi, keeping the luggage in it, and do sightseeing in the morning.

Brilliant! Thank you so much for all the help, LuisJP! :) :) :)